The null hypothesis assumes no difference between groups .Testing Data normality revealed non non-normal distribution so I opted for the Mann Whitney , but testing for Levene Homogeneity of Variance and Box plots showed unequal distribution . does that mean I should accept the null ? or do I have to proceed with other test ? 
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$\begingroup$ Also the log transformation failed to correct the probelm $\endgroup$– MaieMay 1, 2022 at 8:06
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1$\begingroup$ The Wilcoxon Mann Whitney test tests the null hypothesis that the distributions are the same. The alternative hypothesis is that the distributions are not the same. You've already examined the distributions and they don't "look" the same so there's good indication that the null will be rejected. This doesn't make the test invalid. $\endgroup$– dipetkovMay 1, 2022 at 8:30
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1$\begingroup$ However, you have three groups, so should use the Kruskal Wallis test instead, which generalizes the Wilcoxon Mann Whitney test to more than two groups. $\endgroup$– dipetkovMay 1, 2022 at 8:31
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$\begingroup$ I was meant to ask about the fact that there are three groups as well. Of course Maie may be interested in differences between two specific groups but in fact @dipetkov is right, Wilcoxon MW is for comparing two groups. $\endgroup$– Christian HennigMay 1, 2022 at 8:42
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2$\begingroup$ To be very explicit: The rank sum tests WMW and Kruskal-Wallis do not assume that the groups have equal variance or shape. You seem to confuse the assumptions of a test with its null hypothesis. Assumptions: set of conditions that have to be (approximately) true to apply the test. Null hypothesis: if the test is applicable, a statement that the test rejects or doesn't reject based on the evidence in the data. $\endgroup$– dipetkovMay 1, 2022 at 9:44
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1 Answer
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(1) "Accepting" the null is misleading wording anyway (despite being used all over the place) as not rejecting a null hypothesis never means that it is true anyway.
(2) To some extent this depends on what you are interested in. The Wilcoxon MW test will compare rank sums between two groups. The distribution of the test statistic is computed assuming that both distributions are the same, but null hypotheses are never perfectly fulfilled anyway. So you are well "within your rights" to use the test and interpret the result as follows:
Reject: Rank sums are significantly more different than to be expected if distributions were equal, so there is evidence that one distribution tends to produce larger values than the other.
Not reject: The rank sum difference between the groups is not clearly different from what would be expected under equal distributions, so there is no evidence that one distribution produces systematically larger values than the other. This does not mean that distributions are the same, neither does it provide evidence in favour of this. In fact we can see that distributions are not the same, however the test result shows that despite an obvious difference in variation, there may not be any clear difference regarding which distribution generally tends to produce larger values.
So if you're interested in making a statement about whether one group generally produces larger values than the other (in terms of rank sums), and you are not interested in differences in variation, I'd say you can still use the test, although it may be somewhat risky as the way I am putting things here seems nonstandard in the literature and some may find it dodgy.
PS: This generalises to the Kruskal Wallis test for more than two groups.
PPS: As rightly pointed out in a comment, in case of outliers in the data Wilcoxon WM is usually preferable to a t-test based on normality, because the former gives the outliers less influence, same Kruskal-Wallis.
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$\begingroup$ Will you consider mentioning that the rank tests are robust to outliers? I see a few points being labeled in the plot and I suspect that might be in preparation of labeling them "outliers". $\endgroup$– dipetkovMay 1, 2022 at 8:50
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$\begingroup$ Thank you All, so I can disregard the assumption of both WMW and Kruskal-Wallis that groups should be of equal variance or shape ,, and reject or not reject the test result based on the presnece of significant difference in sum of ranks? Also speaking of outliers, does it make median test a better test here? $\endgroup$– MaieMay 1, 2022 at 9:20
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$\begingroup$ @Maie: I'd prefer Wilcoxon MW to median test; the former uses more information. Generally, tests do not reject "test results" but models. See my answer for how to interpret results. If you are interested in other aspects of the distribution than rank sums (such as variances, see my answer), Wilcoxon MW will not do the job. $\endgroup$ May 1, 2022 at 10:03
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